The head of the research project "Axial algebras and related groups" Ilya B. Gorshkov
was awarded the degree of Doctor of Physics and Mathematics. The defense of his dissertation took place at the end of 2020 and the decree to award the degree of Doctor of Sciences was issued at the beginning of April 2021. The dissertation was prepared at the Sobolev Institute of Mathematics SB RAS.
The thesis is devoted to the structure of finite groups with given sizes of conjugate classes. The problem of describing finite groups using properties that can be represented in the form of certain numerical characteristics has a prominent place in group theory. The most frequently used numerical characteristics of groups are the order of the group, the orders of its elements, the orders and indices of various subgroups, the sizes of the classes of conjugate elements. The structure of finite groups with constraints on the size of conjugacy classes has been studied for many years. Thus, back in 1904, in his famous work, W. Burnside showed that a finite group of non-prime order with a conjugacy class whose size is a prime number cannot be simple.
In the dissertation of Ilya Gorshkov, finite groups with given sets of sizes of conjugacy classes and sets of orders of elements were studied. The most significant result of the dissertation is the completion of the verification of Thompson's hypothesis, the question of the validity of which was posed more than 30 years ago: after the announcement of the completion of the classification of finite simple groups, a natural hypothesis arose, which was first expressed by J. Thompson in a letter to W. Shi in 1987, and which in 1992 was recorded in the "Kourovka Notebook" under the number 12.38. In the dissertation, a significant contribution was made to the theory of finite symmetric permutation groups — it was shown that any such group of degree greater than 10 is uniquely characterized in the class of all finite groups by its spectrum.
Ilya's scientific adviser was Doctor of Physics and Mathematics, Professor Andrey V. Vasiliev.